Question

What is the number of distinct triangles with integral valued sides and perimeter as 14?

Answer 1

We know that the sum of any two sides of a triangle is greater than the third side. ....(1)

Here the perimeter of the triangle is $14$ units.

Let divide $14$ into $3$ parts, so that $a+b+c=14$, when $a,b$ and $c$ are integers.

To comply with (1), the sum of any two of $a,b$ or $c$ should be greater than $\frac{14}{2}=7$.

Such possible combinations are

(i) $2+6+6$

(ii) $3+5+6$

(iii) $4+5+5$

(iv) $4+6+4$

No other combination is possible.

So, number of distinct triangles are $4$.